Quillen Notebooks

Daniel Quillen, one of the greatest mathematicians of the latter part of the twentieth century, passed away in 2011 after suffering from Alzheimer’s. For an appreciation of his work and an explanation of its significance, a good place to start is Graeme Segal’s obituary notice, and there’s also quite a bit of material in the AMS Notices.

It’s very exciting to see that the Clay Math Institute now has a project to make available Quillen’s research notebooks. Segal and Glenys Luke have been working on cataloging the set, producing lists of contents of the notebooks, so far from the earliest ones in 1970 up to 1977. Quillen’s work ranged very widely, and for much of the 1980s he was very much involved in what was going on at the boundary of mathematics and quantum field theory. His work on the Mathai-Quillen form provided a beautiful expression for the Thom class of a vector bundle using exactly ingredients that formally generalize to the infinite-dimensional case, where this provides a wonderful way of understanding certain topological quantum field theories. The Mathai-Quillen paper is here, see here for a long expository account of the uses of this in TQFT.

I’ve just started to take a look through the notebooks, and this is pretty mind-blowing. The Mathai-Quillen paper is not the most readable thing in the world; it’s dense with ideas, with motivation and details often getting little attention. Reading the Quillen notebooks is quite the opposite, with details and motivation at the forefront. I just started with Quillen’s notes from Oct. 15 – Nov. 13, 1984, in which he is working out some parts of what appeared later in Mathai-Quillen. This is just wonderful material.

10 Responses to Quillen Notebooks

What a nice coincidence, I just started reading “Homotopical Algebra”!

More generally, these notes are an absolute goldmine for people like me, who are interested in the philosophy of mathematics but not working mathematicians. So much of the process of creating mathematics gets lost in the preparation of mathematical papers: to an outsider, it seems like proofsm theorems and definitions appear almost magically in the mathematician’s mind. Reading notes helps us see that they are mostly just hard working but usually smart people.

Btw, this is my first comment here, although I’ve been following the blog for a couple of years. Thanks Peter for keeping up the good work!

Igor Khavkine,
If you look at the notebooks from the late 1970s, Quillen did return to analysis. I’d also be curious to know about the shift to topology and algebra from his thesis, although perhaps what is surprising is that as a student of Bott he was working on PDEs at all.

Peter, it’s worth noting that not all work on PDEs is in analysis. In fact, Quillen’s work was on the so-called formal theory of PDEs, which is mostly geometric and right around the time of the early 1960s it received a healthy dose of homological algebra, under the influence of the ideas of Donald Spencer. In his thesis, Quillen exploited the homological algebra ideas of Spencer to show that any overdetermined linear differential operator fits into a formally exact sequence of differential operators, each operator completely characterizing the integrability conditions of the preceding one, which is sometimes called the compatibility complex. He also showed some technical results on the ellipticity of this complex. This result is quite important in the formal theory, though apparently it was known much earlier. However, earlier proofs relied on non-geometric, heavily coordinate dependent methods that were developed by Janet and Riquier. Quillen’s solution was on the other hand coordinate independent and fully geometric, in line with the spirit in which the ideas of Spencer were introduced.

I’m not sure of the history of Bott’s interest in the formal theory of PDEs, but it is recorded in his Notes on the Spencer resolution which were circulated around 1963.

I’m curious about why you ask why Quillen abandoned the formal theory of overdetermined systems of PDE’s. Do you use this formal theory yourself?

My view is that he shifted to new directions (which involve similar formal structures) that had a more profound impact on mathematics. In fact, many of the people who worked on this back then, including Spencer, Quillen, Guillemin, Sternberg, Goldschmidt, did their best work on other subjects. It seems to me that the modern formal theory of PDE’s has had relatively little impact on other areas of mathematics.
As far as I know, only Robert Bryant and his students, who use the classical approach of E. Cartan have been able to use the formal theory effectively, usually in applications to differential geometry.

It is indeed a little mysterious why Quillen worked on the formal theory of PDE’s for his thesis. Around that time, Guillemin, Sternberg, and Quillen were around and working on such things. But it seems like Sternberg would have been the more natural choice as an advisor for this topic.

Yes, mysteries all around. You say: “It seems to me that the modern formal theory of PDE’s has had relatively little impact on other areas of mathematics.” In a way you are right, but more is the pity! There are a few, not totally connected, islands where it has been continuously applied. I’d say that much of the potential utility and the power of the formal theory still remains untapped. I for instance have been finding more uses for it (in the mathematical structure of field theory, for instance) the more I’ve learned about it, including the results due to Quillen.

Firstly I think that “Quillen” (my Dad) would be very pleased that you are enjoying what he called his “diary”. He was always trying to get us kids to write diaries, he spoke to us of Samuel Pepys, and it seems, like him, that Dad left something meaningful behind.

I thought I’d add a little info as to how he worked for the benefit of Staffan Angere (and anyone else interested). Certainly in the years of my life (I was born in 1980) Dad did two types of writing. The first was day to day writing which he did on scrap paper, notebooks etc, and the second was the “diary”. The diary was where he would write up, in a more organized way, his thoughts and what he was working on.

It’s reassuring to know that the notebooks are a second, more polished summary of your father’s day to day work. It was kind of frightening to contemplate the possibility that he was directly writing out texts of this high quality without going through a much more confused stage of the sort where the rest of us spend most (if not all…) of our time.